Abstract group $(C_4\times C_{12}).D_4$

• Label: 384.39
• Order: \( 2^{7} \cdot 3 \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\mathrm{Aut}(G)}$: \( 2^{9} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $19$
• Trans deg.: $48$
• Rank: $2$

Group information

Description:	$(C_4\times C_{12}).D_4$
Order:	\(384\)\(\medspace = 2^{7} \cdot 3 \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_2\wr D_4\times D_6$, of order \(1536\)\(\medspace = 2^{9} \cdot 3 \) (generators)
Outer automorphisms:	$C_2^3$, of order \(8\)\(\medspace = 2^{3} \)
Composition factors:	$C_2$ x 7, $C_3$
Derived length:	$3$

This group is nonabelian, supersolvable (hence solvable and monomial), and hyperelementary for $p = 2$.

Group statistics

Order	1	2	3	4	6	8	12	24
Elements	1	27	2	52	6	176	56	64	384
Conjugacy classes	1	3	1	6	3	7	7	8	36
Divisions	1	3	1	6	2	3	5	1	22
Autjugacy classes	1	3	1	6	2	3	5	1	22

Dimension	1	2	4	8	16
Irr. complex chars.	8	10	17	1	0	36
Irr. rational chars.	4	6	6	4	2	22

Minimal Presentations

Permutation degree:	$19$
Transitive degree:	$48$
Rank:	$2$
Inequivalent generating pairs:	$24$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	8	16
Arbitrary	4	8	10

Constructions

Presentation:	$\langle a, b, c, d \mid a^{2}=c^{12}=d^{4}=[a,d]=[c,d]=1, b^{4}=d^{2}, b^{a}=b^{3}c^{3}d, c^{a}=c^{11}d, c^{b}=c^{5}d^{3}, d^{b}=c^{6}d^{3} \rangle$
Permutation group:	Degree $19$ $\langle(1,2,3,7,5,9,12,16)(4,8,14,6,15,13,11,10)(18,19), (1,2)(3,10)(4,13)(5,9) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_4.Q_{16})$ $\,\rtimes\,$ $S_3$	$(C_{12}.Q_{16})$ $\,\rtimes\,$ $C_2$	$C_3$ $\,\rtimes\,$ $(C_4^2.D_4)$	$((C_4\times C_{12}).C_4)$ $\,\rtimes\,$ $C_2$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(Q_8.D_6)$ . $C_4$	$(C_6\times Q_8)$ . $D_4$	$(C_4:Q_8)$ . $D_6$	$(C_2\times Q_8)$ . $D_{12}$	all 17

Elements of the group are displayed as words in the generators from the presentation given above.

Homology

Abelianization:	$C_{2} \times C_{4} $
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 494 subgroups in 106 conjugacy classes, 23 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_2^3.D_{12}$
Commutator:	$G' \simeq$ $C_6\times Q_8$	$G/G' \simeq$ $C_2\times C_4$
Frattini:	$\Phi \simeq$ $C_4:Q_8$	$G/\Phi \simeq$ $D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_{12}.Q_{16}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $(C_4\times C_{12}).D_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_6$	$G/\operatorname{soc} \simeq$ $C_2\wr C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4^2.D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

Normal subgroups up to automorphism

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Subgroup information Click on a subgroup in the diagram to see information about it.

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Classes of subgroups up to conjugation

Order 384: $(C_4\times C_{12}).D_4$

Order 192: $D_{12}.D_4$, $C_{12}.Q_{16}$, $(C_4\times C_{12}).C_4$

Order 128: $C_4^2.D_4$

Order 96: $C_{12}.D_4$ x 2, $C_4:C_{24}$, $Q_8.D_6$, $C_{12}:Q_8$, $C_{12}.D_4$, $D_{12}:C_4$

Order 64: $C_4^2.C_4$, $C_4.Q_{16}$, $D_4.D_4$

Order 48: $C_6\times Q_8$ x 2, $D_{12}:C_2$ x 2, $C_3:\OD_{16}$ x 2, $D_{12}:C_2$, $S_3\times Q_8$, $C_2\times C_{24}$, $C_4:C_{12}$, $C_4\times C_{12}$, $C_3:Q_{16}$, $Q_8:S_3$

Order 32: $C_4.D_4$ x 2, $C_4.C_2^3$, $Q_{16}:C_2$, $C_4:Q_8$, $C_4:C_8$, $C_4\wr C_2$

Order 24: $C_2\times C_{12}$ x 4, $C_4\times S_3$ x 3, $D_{12}$ x 2, $C_3:Q_8$ x 2, $C_3\times Q_8$ x 2, $C_3:C_8$ x 2, $C_3:D_4$, $C_{24}$

Order 16: $D_4:C_2$ x 3, $C_2\times Q_8$ x 3, $\OD_{16}$ x 2, $Q_{16}$, $\SD_{16}$, $C_2\times C_8$, $C_4:C_4$, $C_4^2$

Order 12: $C_{12}$ x 5, $C_2\times C_6$, $D_6$, $C_3:C_4$

Order 8: $C_2\times C_4$ x 7, $Q_8$ x 4, $D_4$ x 3, $C_8$ x 3

Order 6: $C_6$ x 2, $S_3$

Order 4: $C_4$ x 6, $C_2^2$ x 2

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 384: $(C_4\times C_{12}).D_4$

Order 192: $D_{12}.D_4$, $C_{12}.Q_{16}$, $(C_4\times C_{12}).C_4$

Order 128: $C_4^2.D_4$

Order 96: $C_{12}.D_4$ x 2, $C_4:C_{24}$, $Q_8.D_6$, $C_{12}:Q_8$, $C_{12}.D_4$, $D_{12}:C_4$

Order 64: $C_4^2.C_4$, $C_4.Q_{16}$, $D_4.D_4$

Order 48: $C_6\times Q_8$ x 2, $D_{12}:C_2$ x 2, $C_3:\OD_{16}$ x 2, $D_{12}:C_2$, $S_3\times Q_8$, $C_2\times C_{24}$, $C_4:C_{12}$, $C_4\times C_{12}$, $C_3:Q_{16}$, $Q_8:S_3$

Order 32: $C_4.D_4$ x 2, $C_4.C_2^3$, $Q_{16}:C_2$, $C_4:Q_8$, $C_4:C_8$, $C_4\wr C_2$

Order 24: $C_2\times C_{12}$ x 4, $D_{12}$ x 2, $C_4\times S_3$ x 2, $C_3:Q_8$ x 2, $C_3\times Q_8$ x 2, $C_3:C_8$ x 2, $C_3:D_4$, $C_{24}$

Order 16: $D_4:C_2$ x 3, $C_2\times Q_8$ x 3, $\OD_{16}$ x 2, $Q_{16}$, $\SD_{16}$, $C_2\times C_8$, $C_4:C_4$, $C_4^2$

Order 12: $C_{12}$ x 5, $C_2\times C_6$, $D_6$, $C_3:C_4$

Order 8: $C_2\times C_4$ x 6, $Q_8$ x 4, $D_4$ x 3, $C_8$ x 3

Order 6: $C_6$ x 2, $S_3$

Order 4: $C_4$ x 6, $C_2^2$ x 2

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 384: $(C_4\times C_{12}).D_4$ ($C_1$)

Order 192: $D_{12}.D_4$ ($C_2$), $C_{12}.Q_{16}$ ($C_2$), $(C_4\times C_{12}).C_4$ ($C_2$)

Order 96: $C_{12}.D_4$ ($C_4$), $Q_8.D_6$ ($C_4$), $C_{12}:Q_8$ ($C_2^2$)

Order 64: $C_4.Q_{16}$ ($S_3$)

Order 48: $C_6\times Q_8$ ($D_4$), $C_6\times Q_8$ ($C_2\times C_4$), $C_4\times C_{12}$ ($D_4$)

Order 32: $C_4:Q_8$ ($D_6$)

Order 24: $C_2\times C_{12}$ ($C_2^2:C_4$)

Order 16: $C_4^2$ ($C_3:D_4$), $C_2\times Q_8$ ($D_{12}$), $C_2\times Q_8$ ($C_4\times S_3$)

Order 12: $C_2\times C_6$ ($C_2^3:C_4$)

Order 8: $C_2\times C_4$ ($D_6:C_4$)

Order 6: $C_6$ ($C_2\wr C_4$)

Order 4: $C_2^2$ ($C_2^2.D_{12}$)

Order 3: $C_3$ ($C_4^2.D_4$)

Order 2: $C_2$ ($C_2^3.D_{12}$)

Order 1: $C_1$ ($(C_4\times C_{12}).D_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 384: $(C_4\times C_{12}).D_4$ ($C_1$)

Order 192: $D_{12}.D_4$ ($C_2$), $C_{12}.Q_{16}$ ($C_2$), $(C_4\times C_{12}).C_4$ ($C_2$)

Order 96: $C_{12}.D_4$ ($C_4$), $Q_8.D_6$ ($C_4$), $C_{12}:Q_8$ ($C_2^2$)

Order 64: $C_4.Q_{16}$ ($S_3$)

Order 48: $C_6\times Q_8$ ($D_4$), $C_6\times Q_8$ ($C_2\times C_4$), $C_4\times C_{12}$ ($D_4$)

Order 32: $C_4:Q_8$ ($D_6$)

Order 24: $C_2\times C_{12}$ ($C_2^2:C_4$)

Order 16: $C_4^2$ ($C_3:D_4$), $C_2\times Q_8$ ($D_{12}$), $C_2\times Q_8$ ($C_4\times S_3$)

Order 12: $C_2\times C_6$ ($C_2^3:C_4$)

Order 8: $C_2\times C_4$ ($D_6:C_4$)

Order 6: $C_6$ ($C_2\wr C_4$)

Order 4: $C_2^2$ ($C_2^2.D_{12}$)

Order 3: $C_3$ ($C_4^2.D_4$)

Order 2: $C_2$ ($C_2^3.D_{12}$)

Order 1: $C_1$ ($(C_4\times C_{12}).D_4$)

Series

Derived series

$(C_4\times C_{12}).D_4$

$\rhd$

$C_6\times Q_8$

$\rhd$

$C_2$

$\rhd$

$C_1$

Chief series

$(C_4\times C_{12}).D_4$

$\rhd$

$D_{12}.D_4$

$\rhd$

$C_{12}:Q_8$

$\rhd$

$C_6\times Q_8$

$\rhd$

$C_2\times C_{12}$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$(C_4\times C_{12}).D_4$

$\rhd$

$C_6\times Q_8$

$\rhd$

$C_2\times C_{12}$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_6$

$\rhd$

$C_3$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2^2$

$\lhd$

$C_2\times C_4$

$\lhd$

$C_4:Q_8$

$\lhd$

$C_4.Q_{16}$

Character theory

Complex character table

See the $36 \times 36$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $22 \times 22$ rational character table.